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''four-functions inequality''

An inequality in which an inequality for four functions on a finite distributive lattice applies also to additive extensions of the functions on lattice subsets. Let  $  ( \Gamma, \prec ) $
be a finite [[Distributive lattice|distributive lattice]] (see also [[FKG inequality|FKG inequality]]), such as the power set of a finite set ordered by proper inclusion. For subsets  $  A $, 
$  B $
of  $  \Gamma $, 
define  $  A \lor B = \{ {a \lor b } : {a \in A, b \in B } \} $
and  $  A \wedge B = \{ {a \wedge b } : {a \in A, b \in B } \} $. 
If  $  A $
or  $  B $
is empty,  $  A \lor B = A \wedge B = \emptyset $. 
Given  $  f : \Gamma \rightarrow \mathbf R $, 
let  $  f ( A ) = \sum _ {a \in A }  f ( a ) $.

The Ahlswede–Daykin inequality says that if  $  f _ {1} $, 
$  f _ {2} $, 
$  f _ {3} $, 
and  $  f _ {4} $
map  $  \Gamma $
into  $  [ 0, \infty ) $
such that

$$ 
f _ {1} ( a ) f _ {2} ( b ) \leq  f _ {3} ( a \lor b ) f _ {4} ( a \wedge b )  \textrm{ for  all  }  a, b \in \Gamma,
$$

then

$$ 
f _ {1} ( A ) f _ {2} ( B ) \leq  f _ {3} ( A \lor B ) f _ {4} ( A \wedge B )  \textrm{ for  all  }  A, B \subseteq \Gamma .
$$

See [[#References|[a1]]] or [[#References|[a2]]], [[#References|[a4]]], [[#References|[a7]]] for a proof.

The inequality is very basic and is used in proofs of other inequalities (cf. [[#References|[a2]]], [[#References|[a3]]], [[#References|[a4]]], [[#References|[a5]]], [[#References|[a7]]]), including the [[FKG inequality|FKG inequality]] [[#References|[a6]]] and the [[Fishburn–Shepp inequality|Fishburn–Shepp inequality]] [[#References|[a3]]], [[#References|[a8]]].

See also [[Correlation inequalities|Correlation inequalities]]; [[Holley inequality|Holley inequality]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Ahlswede,   D.E. Daykin,   "An inequality for the weights of two families, their unions and intersections"  ''Z. Wahrscheinlichkeitsth. verw. Gebiete'' , '''43'''  (1978)  pp. 183–185</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  B. Bollobás,   "Combinatorics" , Cambridge Univ. Press  (1986)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  P.C. Fishburn,   "A correlational inequality for linear extensions of a poset"  ''Order'' , '''1'''  (1984)  pp. 127–137</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  P.C. Fishburn,   "Correlation in partially ordered sets"  ''Discrete Appl. Math.'' , '''39'''  (1992)  pp. 173–191</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  P.C. Fishburn,   P.G. Doyle,   L.A. Shepp,   "The match set of a random permutation has the FKG property"  ''Ann. of Probab.'' , '''16'''  (1988)  pp. 1194–1214</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  C.M. Fortuin,   P.N. Kasteleyn,   J. Ginibre,   "Correlation inequalities for some partially ordered sets"  ''Comm. Math. Phys.'' , '''22'''  (1971)  pp. 89–103</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  R.L. Graham,   "Applications of the FKG inequality and its relatives" , ''Proc. 12th Internat. Symp. Math. Programming'' , Springer  (1983)  pp. 115–131</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  L.A. Shepp,   "The XYZ conjecture and the FKG inequality"  ''Ann. of Probab.'' , '''10'''  (1982)  pp. 824–827</TD></TR></table>